#413901

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1489 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,12)(8,20)(9,14)(10,24)(11,15)(13,23)(16,25)(17,31)(18,27)(19,28)(21,35)(22,29)(26,30)(32,34)(33,36);(2,4)(5,7)(6,12)(8,17)(9,16)(10,18)(11,19)(13,26)(14,25)(15,28)(20,34)(21,32)(22,30)(23,33)(24,27)(29,36)(31,35);(7,16)(9,10)(12,25)(14,24)(17,30)(18,19)(20,35)(22,23)(26,34)(27,28)(31,36)(32,33);(5,8)(6,13,15,29)(7,17)(9,22)(10,23)(11,21)(12,26,28,36)(14,20,24,35)(16,30)(18,33)(19,32)(25,34,27,31)

Polycyclic presentation:

2^9|d^2=gh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256516, #72256519, #72256520
maximal normal subgroup of	#72256516, #72256519, #72256520

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321097);